There are various levels of protection; if you file corporate papers I don't think any other corporation in that state can have the same name. Otherwise, there is no protection, really. You could I suppose file for trademark protection (expensive and lengthy I think) and then if someone else used the name, you could issue a cease and desist letter, and if the other barn continued to use the name you could sue if there would be confusion.
All the above is highly theoretical -- no one trademarks barn names!
If you have a private farm, why do you care? I ask because perhaps whatever your concern is could be addressed in other ways.
Agree w/ the above post. I would venture to say there are plenty of Stonebridge, Stoneridge, Fox Hollow, River View etc type names all over the place - most people names their farms/barns after something that means something to them - whether it's a place, person, animal, etc. Even companies run the risk of having other businesses w/ similar names - logos are trademarked, I don't know that you can trademark a name per se perhaps copywrited but then you'd run the risk of having to do a search and potential suit if you trademarked or copywrited a name. for instance ages ago there used to be a small publication Polo Magazine - well seems that Ralph Lauren didn't like the fact that there was a magazine called Polo (which zip to do Lauren's clotheshis clothes) so Lauren threatened to sue the owner of Polo Mag... w/a relatively small subscribership the mag relented to Lauren... which is really rich considering Lauren made/is making a bundle off the Sport